MIT OpenCourseWare http://ocw.mit.edu Electromechanical Dynamics For any use or distribution of this textbook, please cite as follows: Woodson, Herbert H., and James R. Melcher. Electromechanical Dynamics. 3 vols. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsAppendix ESUMMARY OF PARTS I AND IIAND USEFUL THEOREMSIDENTITIESAx B.C= AB x C,A x (B x C) = B(A. C) -C(A .B)V(o + y) = V + Vy,V. (A + B) = V A + V B,V x (A + B)= V x A + V x B,V(ov) = V P + v vo,V-(VA)= A -VyB + V .A,V.(A x B)= B.V x A -A V x B,v. v-= Vý,V-V x A= 0,V x V =0,V x (V x A) = V(V .A) -V2A,(V x A) x A = (A .V)A -lV(A .A),V(A .B)= (A .V)B + (B -V)A + A x (V x B) + B x (V x A)V x (OA)= Vo x A + OV x A,V x (A x B) = A(V .B) -B(V .A) + (B .V)A -(A .V)B.ElTHEOREMSbibVýk -Al = rDivergence theoremsA .nda=JV.AdVStokes's theorem AA dl =(V x A) .n daý,C S(Vx ~daITable 1.2 Summary of Quasi-Static Electromagnetic EquationsDifferential EquationsIntegral EquationsMagnetic field systemElectric field systemV x H = J,V. J= 0aBVXE= -TtVX E =0V.D= PfV.J= ataDVXH=J1+(1.1.1)(1.1.11)(1.1.12)(1.1.14)(1.1.15)H -dl = J" -n daBB.nda = 0SJ, -n da = 0E'.dl =-f B .n dawhere E' = E + v X BEgE.dl= 0DD'nda=fvpp dVJf'-nda d vpdVH'.dlT = J .n da + D -n dawhere J' = J= -pfvH'=H-v XD(1.1.20)(1.1.21)(1.1.22)(1.1.23)(1.1.24)(1.1.25)(1.1.26)(1.1.27)Table 2.1 Summary of Terminal Variables and Terminal RelationsMagnetic field system Electric field system-VCDefinition of Terminal VariablesFlux ChargeA = B.nda qk= f pdVCurrent Voltageik ý f -n'da Vk = fE diTerminal ConditionsdAk dqkSdt dtiA, = 4.(i ... iN; geometry) qk = qk(v1• • • vv; geometry)i, = ik(.1AN; geometry) vk = vk(ql1" "qN; geometry)Table 3.1 Energy Relations for an Electromechanical Coupling Network with N Electricaland M Mechanical Terminal Pairs*Magnetic Field Systems Electric Field SystemsConservation of EnergyN 31 N 31dWm = J ij dAj - f e dxj (a) dWe -= v dq, -' fe d-j (b)j=1 j=1 j=1 j=1N M NV MdW2 = I A di, + I fe dx- (c) dWi = I qj dv + I y L dx3 (d)3=1 i=1 j=1 t=1Forces of Electric Origin,j = 1, ... Me= -a ANl. Ax ...) (e-) hf = aWe(ql.... q; x1. .x) (f)axj x(e jef M 1t) ((a ....._ VN;l ...X1, x,1)at(i .x. (g) he =. (h)Relation of Energy to CoenergyN NWm + w" = ij (i) We + We' = =vq (j)3=1 3=1Energy and Coenergy from Electrical Terminal RelationsWm=. d (k) We = v(ql. . , q 0 ; x1 . x) dq ()j= ( , 2'0 0;4) ( W f- ,q,0,...10;xj,...m)dq (1)N i N P'tWm i,... ij-1, 1, 0i,...,0; xI,.... x) di (mi ) W = q(vl,... I, vj1', 0 ...0; x...x) dv (n)j=1 0 o=1* The mechanical variables f and x, can be regarded as thejth force and displacement or thejth torque T, and angular displacement Q0.Table 6.1 Differential Equations, Transformations, and Boundary Conditions for Quasi-static Electromagnetic Systems withMoving MediaDifferential Equations Transformations Boundary ConditionsV x H = J (1.1.1) H' = H (6.1.35) n X(Ha -Hb) = K (6.2.14)V. B =O0 (1.1.2) B' = B (6.1.37) n.(Ba --Bb) = 0 (6.2.7)Magnetic V. = 0 (1.1.3) J = (6.1.36) n.(J, -_Jfb) + V, . K7 = 0 (6.2.9)systems aBVx E = -(1.1.5) E' = E + vr x B (6.1.38) n X (Ea -Eb) = v,(Ba-Bb) (6.2.22)B = p0(H + M) (1.1.4) M' = M (6.1.39)V x E = 0 (1.1.11) E' = E (6.1.54) n X (Ea -Eb) = 0 (6.2.31)V.D= p, (1.1.12) D' = D (6.1.55) n.(Da -Db)=rr (6.2.33)P = Pf (6.1.56)Electric . J = -a (1.1.14) J = Jf -pfvr (6.1.58) n* (Jfa -_ Jb) + V.,-Kf = V((pfa fb) _ " (6.2.36)field atsystems aDV X H = Jf+ -(1.1.15) H'= H -v x D (6.1.57) n X (Ha - Hb) = K, + van X [n x (Da -Db)] (6.2.38)D = eoE + P (1.1.13) P' = P
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